8 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



triple integral by which the value of V" is expressed, is the same as 

 the one before given for V, except that the integration relative to r, 

 instead of extending from /=0 to r'=l, ought only to extend from 

 r =r to r= 1. 



But the general term in the function J'{r'^) being represented by 

 Atr'^\ the part of T^ dependent on this term will evidently be 



(2) Atr'fde'dw' sin 9'.QJr'''+^-^-''dr' {l-r'y-, 



the limits of the integrals being the same as before. 



We thus see that the value of T, and consequently of F'" would 

 immediately be obtained, provided we had the value of the general 



integral 



flr^dril-ry, 



which being expanded and integrated becomes 



^ 1 +Mzl). 1 _&c. 



b + l l'b + 3 1.2 'b + 5 



^+1 Q fj,+3 /3(/3-l) r'+» 



+ T • i — ;r — ~ 7 r. • ~i = + <^c- 



6 + 1 16 + 3 1.2 '6 + 5 



but since the first line of this expression is the well known expansion of 



(f) 



r lf\ r li 



or 



nT 



m' 



when n = 2.p — h + \ and 5' = 2(/3 + l) we have ultimately, 



By means of the result here obtained, we shall readily find the 

 value of the expression (2) which will evidently contain one term multi- 

 plied by r' and an infinite number of others, in all of which the quantity 

 r is affected with the exponent n. But as in the case under considera- 

 tion, n may represent any number whatever, fractionary or irrational, 



